WOMAN: Exercise 7.
Find the constant term in the expansion
of (x + 1 divided by x squared) all to the power of 6.
The general term in the expansion of (a + b) all to the power of n
is n choose r times a to the power of n minus r
times b to the power of r, where r is between 0 and n inclusive.
So in this particular expansion
the general term will be 6 choose r times x to the power of 6 minus r
times 1 divided by x squared all to the power of r,
where r is between 0 and 6 inclusive.
So considering this general term, we need to work out
which value of r is going to give us the constant term,
and we acknowledge that the constant term will occur
when the 'x's in the general term cancel out.
And in this case that's going to happen when r is equal to 2,
which will give 6 choose 2 x to the power of 4
and 1 divided by x squared all squared,
which, when we tidy that up a little,
gives us 15x to the power of 4 times 1 over x to the power of 4.
So x to the power of 4 is going to cancel out,
leaving us with a constant term of 15.